Further and deeper exploration of paradoxes and challenges of intuition and logic can be found in my recently published book, Probability, Choice and Reason.

Ask someone to toss a fair coin 32 times. Which of the following rows of coin toss patterns is more likely to result if they actually do toss the coins and record them accurately, and which is likely to be the fake?

HTTHTHTTHHTHTHHTTTHTHTTHTHHTTHHT

OR

HTTHTHTTTTTHTHTTHHHHTTHTHTHHTHHT

In both cases, there are 15 heads and 17 tails.

But would we expect a run (r) of five Heads or a run of five tails in the series, where r is the length of the run?

The chance of five heads = (1/2) to the power of r = (1/2) to the power of 5 = 1/32. But there are 28 opportunities for a run of five heads in 32 tosses. Same for a run of five tails.

A good rule of thumb is that when N (the number of opportunities for a run to take place) x (1/2 to the power of r) equals 1, it is likely that a run of length, r, will appear in the sequence. So, a run of length r is likely to appear when N = 2 to the power of r.

In the case of 32 coin tosses, with 28 possible runs of length five, N (28) is almost equal to 2 to the power of 5 (32). So a run of five heads (or of tails) is likely if a fair coin is tossed randomly 32 times in a row, and a run of four is almost certain.

Now look at the series of coin tosses above. The first series of 32 coin tosses has no run of heads (or tails) longer than three. The second series has a run of five tails and of four heads.

It is very likely indeed, therefore, that the second series is the genuine one, and the first one is the fake.

Appendix

Probability of 5 heads in a row = 1/32.

Probability of NOT getting 5 heads in a row from a particular run of 5 coin tosses = 31/32

Chance of NOT getting 5 heads in a row from 28 runs of five coin tosses = (31/32) to the power of 28 = 41.1%.

Therefore, the probability of getting 5 heads in a row from 28 runs of five coin tosses = 58.9%.

Similarly for tails.

The Probability of 5 heads OR 5 tails in a row = 1/32 + 1/32 = 1/16

Probability of NOT getting 5 heads OR 5 tails in a row from a particular run of 5 coin tosses = 15/16

Chance of NOT getting 5 heads OR 5 tails in a row from 28 runs of five coin tosses = (15/16) to the power of 28 =16.4%.

Therefore, the probability of getting 5 heads OR 5 tails in a row from 28 runs of five coin tosses = 83.6%

Probability of 4 heads in a row = 1/16.

Probability of NOT getting 4 heads in a row from a particular run of 4 coin tosses = 15/16

Chance of NOT getting 4 heads in a row from 29 runs of four coin tosses = (15/16) to the power of 29 = 15.4%.

Therefore, the probability of getting 5 heads in a row from 28 runs of five coin tosses = 84.6%.

Similarly for tails.

Probability of 4 heads OR 4 tails in a row = 1/16 + 1/16 = 1/8

Probability of NOT getting 4 heads OR 4 tails in a row from a particular run of 4 coin tosses = 7/8

Chance of NOT getting 4 heads OR 4 tails in a row from 29 runs of four coin tosses = (7/8) to the power of 29 = 2.1%

Therefore, the probability of getting 4 heads OR 4 tails in a row from 29 runs of four coin tosses = 97.9%

Exercise

When Nasser Hussain was England cricket captain during 200-01, he lost all 14 coin tosses in the international matches he captained. Given that he captained England in all international matches about a hundred times, what was the probability that he would face this long a losing streak during his captaincy?